Group MMSE-DFD with order and filter computation for reception on a cellular downlink

ABSTRACT

The present method resides in a user destination receiver to exploit the structure of the transmitted signals to design filters that yield improved performance. Moreover, the computational cost of designing these filters can be reduced and the demodulation complexity can be kept low. Further, the present method enables determining the order of decoding the transmitter sources. The present method provides group MMSE decision feedback decoding for the case when all the sources transmit at fixed pre-determined rates and the MCS employed by each source is known to the destination. The present method includes a filtering technique and an order and filter computation process, both improvements over previous efforts at group MMSE decision feedback decoding.

This application claims the benefit of U.S. Provisional Application No.60/894,555, entitled “Analysis of Multiuser Stacked Space-TimeOrthogonal and Quasi-Orthogonal Designs”, filed on Mar. 13, 2007, isrelated to U.S. patent application Ser. No. 12/047,514, entitled “GROUPLMMSE DEMODULATION USING NOISE AND INTERFERENCE COVARIANCE MATRIX FORRECEPTION ON A CELLULAR DOWNLINK”, filed Mar. 13, 2008, related to U.S.patent application Ser. No. 12/047,544, entitled “GROUP MMSE-DFD WITHRATE (SINR) FEEDBACK AND PRE-DETERMINED DECODING ORDER FOR RECEPTION OFA CELLULAR DOWNLINK”, filed Mar. 13, 2008, and related to U.S. patentapplication Ser. No. 12/047,555, entitled “GROUP MMSE-DFD WITH RATE(SINR) FEEDBACK AND WITHOUT PRE-DETERMINED DECODING ORDER FOR RECEPTIONOF A CELLULAR DOWNLINK”, filed Mar. 13, 2008, all of which theircontents are incorporated by reference herein.

BACKGROUND OF THE INVENTION

The present invention relates generally to wireless communications and,more particularly, to a method of group minimum-mean-squared-errordecision-feedback-decoder (MMSE-DFD) with order and filter computationfor reception on a cellular Downlink.

A wireless cellular system consists of several base-stations or accesspoints, each providing signal coverage to a small area known as a cell.Each base-station controls multiple users and allocates resources usingmultiple access methods such as OFDMA, TDMA, CDMA, etc., which ensurethat the mutual interference between users within a cell (a.k.a.intra-cell users) is avoided. On the other hand co-channel interferencecaused by out-of-cell transmissions remains a major impairment.Traditionally cellular wireless networks have dealt with inter-cellinterference by locating co-channel base-stations as far apart aspossible via static frequency reuse planning at the price of loweringspectral efficiency. More sophisticated frequency planning techniquesinclude the fractional frequency reuse scheme, where for the cellinterior a universal reuse is employed, but for the cell-edge the reusefactor is greater than one. Future network evolutions are envisioned tohave smaller cells and employ a universal (or an aggressive) frequencyreuse. Therefore, some sort of proactive inter-cell interferencemitigation is required, especially for edge users. Recently, it has beenshown that system performance can be improved by employing advancedmulti-user detection (MUD) for interference cancellation or suppression.However, in the downlink channel which is expected to be the bottleneckin future cellular systems, only limited signal processing capabilitiesare present at the mobile which puts a hard constraint on thepermissible complexity of such MUD techniques.

In the downlink, transmit diversity techniques are employed to protectthe transmitted information against fades in the propagationenvironment. Future cellular systems such as the 3GPP LTE system arepoised to deploy base-stations with two or four transmit antennas inaddition to legacy single transmit antenna base-stations and cater tomobiles with up to four receive antennas. Consequently, these systemswill have multi-antenna base-stations that employ space-only inner codes(such as long-term beam forming) and space-time (or space-frequency)inner codes based on the 2×2 orthogonal design (a.k.a. Alamouti design)and the 4×4 quasi-orthogonal design, respectively. The aforementionedinner codes are leading candidates for downlink transmit diversity inthe 3GPP LTE system for data as well as control channels. The systemdesigner must ensure that each user receives the signals transmitted onthe control channel with a large enough SINR, in order to guaranteecoverage and a uniform user experience irrespective of its position inthe cell. Inter-cell interference coupled with stringent complexitylimits at the mobile makes these goals significantly harder to achieve,particularly at the cell edge.

The idea of using the structure of the co-channel interference to designfilters has been proposed, where a group decorrelator was designed foran uplink channel with two-users, each employing the Alamouti design asan inner code. There has also been derived an improved groupdecorrelator for a multi-user uplink where each user employs the 4×4quasi-orthogonal design of rate 1 symbol per channel use. Improved groupdecorrelators have resulted in higher diversity orders and have alsopreserved the (quasi-) decoupling property of the constituent (quasi-)orthogonal inner codes.

Accordingly, there is a need for a method of reception on a downlinkchannel with improved interference suppression and cancellation, whichexploits the structure or the spatio-temporal correlation present in theco-channel interference.

SUMMARY OF THE INVENTION

In accordance with the invention, a method for decoding in a wirelessdownlink channel, where all dominant transmitting sources use innercodes from a particular set, including the steps of: estimating channelmatrices seen from all dominant transmitter sources in response to apilot or preamble signal transmitted by each such source; convertingeach estimated channel matrix into an effective channel matrixresponsive to the inner code of the corresponding transmitting source;obtaining the received observations in a linear equivalent form whoseoutput is an equivalent of the received observations and in which theeffective channel matrix corresponding to each dominant transmittingsource inherits the structure of its inner code; i) determining an orderfor processing each of the transmitting sources; ii) computing a filterfor each transmitting source that will be decoded; iii) demodulating anddecoding each transmitting source responsive to the determined orderfrom step i) assuming perfect cancellation of signals of preceding orpreviously decoded transmitting sources; and iv) re-encoding the decodedmessage of each transmitting source, except the source decoded last,responsive to the modulation and coding scheme employed by the sourceand the corresponding effective channel matrix and subtracting it fromthe received observations in the equivalent linear form.

BRIEF DESCRIPTION OF DRAWINGS

These and other advantages of the invention will be apparent to those ofordinary skill in the art by reference to the following detaileddescription and the accompanying drawings.

FIG. 1 is a schematic of adjacent cell interference in a cellularnetwork demonstrating a problem which the invention solves;

FIG. 2 is a receiver flow diagram for the case when all the transmittersources transmit at fixed pre-determined rates and the inner code andthe modulation and coding scheme (MCS) employed by each source are knownto the destination, in accordance with the invention; and

FIG. 3 is a receiver flow diagram for the order and filter computationprocess employed in the group MMSE-DFD process of FIG. 2, in accordancewith the invention.

DETAILED DESCRIPTION

1. Introduction

The invention is directed to a cellular downlink where the user receivesdata from the serving base-station and is interfered by adjacentbase-stations, as shown by the diagram of FIG. 1. In general, theinvention is applicable in a scenario where the user (destination)receives signals simultaneously from multiple sources and is interestedin the signal transmitted by one (desired) source or a few sources ofinterest. The signals transmitted by all base-stations have structure.In particular the inner codes used by all transmitters are from a set ofinner codes [(2)-to-(5)]. The inner code and the modulation and codingscheme of each source is known to the destination.

The inventive method resides in the user (destination) receiver designin which we exploit the structure of the transmitted signals to designfilters that yield improved performance (henceforth referred to asimproved filters). Moreover, the computational cost of designing thesefilters can be reduced (Efficient filter design: see Section 4 below]and the demodulation complexity can be kept low, for example see Theorem1 below. Further, the order of decoding the sources is also determinedby the inventive method [See Section 5 below].

More specifically, the inventive method provides group MMSE decisionfeedback decoding for the case when all the sources transmit at fixedpre-determined rates and the MCS employed by each source is known to thedestination. The inventive method includes a filtering technique and anorder and filter computation process, both improvements over previousefforts at group MMSE decision feedback decoding.

The process steps in accordance with the invention are shown in FIG. 2and FIG. 3. FIG. 2 is a receiver flow diagram for the case when all thetransmitter sources transmit at fixed pre-determined rates and themodulation and coding scheme (MCS) employed by each source is known tothe destination, in accordance with the invention. FIG. 3 is a receiverflow diagram for the order and filter computation process employed inthe group MMSE-DFD process of FIG. 2, in accordance with the invention.

Referring now to FIG. 2, the receiver is intialized 20 with an innercode and modulation and coding scheme (MCS) of each source, and thereceived signal observations determined in accordance with the matrixrelationship (6) (see formula derivation and details in 2. SystemsDescriptions, A. System Model). In response to a pilot or preamblesignal sent from each source, an estimation of the channel matrix ofeach source is performed 21, followed by an order and filter computationprocess 22 (see FIG. 3 for details). In the final step 23, i) thetransmitting sources are processed according to the order determined,ii) each transmitter source is decoded assuming perfect cancellation ofthe signals of preceding transmitter sources, iii) use of the filterobtained and decoupling property (see Theorem 1) for an efficientdemodulation, and iv) for each decoded source (except the source decodedlast), the receiver re-encodes the decoded message and subtracts fromthe received observations.

Referring now to FIG. 3, the order and filter computation is intialized30 with channel matrix estimates of all transmitter sources, a set I ofindices of all transmitter sources of interst, rates of all transmittersources, a set S of indices of all transmitter sources and an empty setO. The process then considers each transmitter source k in set S andselects the one which maximizes a metrik m(k,S\k) (according to formula(41) or (42) or (43)). Supposing that transmitter source selected issource j (metrics can be efficiently computed using Section 4,Theorem 1) 31. Then the process updates S=S\j, O={O,j} and computes thefilter for transmitter source j (see section 4) and stores thecomputation 32. At step 33, If the set O does not contain the set I ofindices of all transmitter sources of interest, then steps 31 and 32 arerepeated. If the set O does contain the set I of indices of alltransmitter sources of interest then the ordered set O and allcorresponding filters are output to the FIG. 2 process 34.

2. System Descriptions

2.1. System Model

We consider a downlink fading channel, depicted in FIG. 1, where thesignals from K base-stations (BSs) are received by the user of interest.The user is equipped with N≧1 receive antennas and is served by only oneBS but interfered by the remaining K−1 others. The BSs are also equippedwith multiple transmit antennas and transmit using any one out of a setof three space-time inner codes. The 4×N channel output received overfour consecutive symbol intervals, is given byY=XH+V,  (1)where the fading channel is modeled by the matrix H. For simplicity, weassume a synchronous model. In practice this assumption is reasonable atthe cell edge and for small cells. Moreover, the model in (1) is alsoobtained over four consecutive tones in the downlink of a broadbandsystem employing OFDM such as the 3GPP LTE system. We partition H asH=[H₁ ^(T), . . . , H_(K) ^(T)]^(T), where H_(k) contains the rows of Hcorresponding to the k^(th) BS. The channel is quasi-static and thematrix H stays constant for 4 symbol periods after which it may jump toan independent value. The random matrix H is not known to thetransmitters (BSs) and the additive noise matrix V has i.i.d. C

(0,2σ²) elements.

The transmitted matrix X can be partitioned as =[x₁, . . . , x_(K)]where

$\begin{matrix}{{X_{k} = \begin{bmatrix}x_{k,1} & x_{k,2} & x_{k,3} & x_{k,4} \\{- x_{k,2}^{\dagger}} & x_{k,1}^{\dagger} & {- x_{k,4}^{\dagger}} & x_{k,3}^{\dagger} \\x_{k,3} & x_{k,4} & x_{k,1} & x_{k,2} \\{- x_{k,4}^{\dagger}} & x_{k,3}^{\dagger} & {- x_{k,2}^{\dagger}} & x_{k,1}^{\dagger}\end{bmatrix}},} & (2)\end{matrix}$when the k^(th) BS employs the quasi orthogonal design as its inner codeand

$\begin{matrix}{{X_{k} = \begin{bmatrix}x_{k,1} & x_{k,2} \\{- x_{k,2}^{\dagger}} & x_{k,1}^{\dagger} \\x_{k,3} & x_{k,4} \\{- x_{k,4}^{\dagger}} & x_{k,3}^{\dagger}\end{bmatrix}},} & (3)\end{matrix}$when the k^(th) BS employs the Alamouti design and finallyX_(k)=[x_(k,1) x_(k,2) x_(k,3) x_(k,4)]^(T).  (4)when the k^(th) BS has only one transmit antenna. The power constraintsare taken to be E{|x_(k,q)|²}≦2w_(k), 1≦k≦K, 1≦q≦4.

We also let the model in (1) include a BS with multiple transmitantennas which employs beamforming. In this caseX_(k)[x_(k,1) x_(k,2) x_(k,3) x_(k,4)]^(T)u_(k),  (5)where u_(k) is the beamforming vector employed by BS k. Note that X_(k)in (5) can be seen as a space-only inner code. Also, the beamforming inwhich vector u_(k) only depends on the long-term channel information, isreferred to as long-term beamforming. We can absorb the vector u_(k)into the channel matrix H_(k) and consider BS k to be a BS with a singlevirtual antenna transmitting (4). Notice that the inner codes in(2)-to-(5) all have a rate of one symbol per-channel-use and we assumethat the desired BS employs any one out of these inner codes.Furthermore, we can also accommodate an interfering BS with multipletransmit antennas transmitting in the spatial multiplexing (a.k.a.BLAST) mode as well as an interfering BS with multiple transmit antennasemploying a higher rank preceding. In such cases, each physical orvirtual transmit antenna of the interfering BS can be regarded as avirtual interfering BS with a single transmit antenna transmitting (4).Then since the codewords transmitted by these virtual BSs areindependent they can be separately decoded when the interferencecancellation receiver is employed.

Let Y_(n) and V_(n) denote the n^(th), 1≦n≦N, columns of the matrices Yand V with Y_(n) ^(R), Y_(n) ^(I) and V_(n) ^(R), V_(n) ^(I) denotingtheir real and imaginary parts, respectively. We define the 8N×1 vectors{tilde over (y)}

[(Y₁ ^(R))^(T), (Y₁ ^(I))^(T), . . . , (Y_(N) ^(R))^(T), (Y_(N)^(I))^(T)]^(T), {tilde over (v)}

[(V₁ ^(R))^(T), (V₁ ^(I))^(T), . . . , (V_(N) ^(R))^(T), (V_(N)^(I))^(T)]^(T).

Then, {tilde over (y)} can be written as{tilde over (y)}={tilde over (H)}{tilde over (x)}+{tilde over (v)},  (6)where {tilde over (x)}

[{tilde over (x)}₁ ^(T), . . . , {tilde over (x)}_(K) ^(T)]^(T) with{tilde over (x)}=[x_(k,1) ^(R), . . . , x_(k,4) ^(R), x_(k,1) ^(I), . .. , x_(k,4) ^(I)]^(T) and {tilde over (H)}=[{tilde over (H)}₁, . . . ,{tilde over (H)}_(K)]=[{tilde over (h)}₁, . . . , {tilde over(h)}_(8K)]. Further when the k^(th) BS employs either thequasi-orthogonal design or the Alamouti design we can expand {tilde over(H)}_(k) as{tilde over (H)} _(k) =[{tilde over (h)} _(8k−7) , . . . , {tilde over(h)} _(8k)]=[(I _(N) {circle around (×)}C ₁){tilde over (h)} _(8k−7), (I_(N) {circle around (×)}C ₂){tilde over (h)} _(8k−7), . . . , (I _(N){circle around (×)}C ₈) {tilde over (h)} _(8k−7)],  (7)where {circle around (×)} denotes the Kronecker product, C₁=I₈ and

C 2 = I 2 ⊗ [ 0 1 0 0 - 1 0 0 0 0 0 0 1 0 0 - 1 0 ] ⁢ ⁢ C 3 = I 2 ⊗ [ 0 01 0 0 0 0 1 1 0 0 0 0 1 0 0 ] C 4 = I 3 ⁡ [ 0 0 0 1 0 0 - 1 0 0 1 0 0 - 10 0 0 ] C 5 = J 2 ⊗ [ 1 0 0 0 0 - 1 0 0 0 0 1 0 0 0 0 - 1 ] ⁢ ⁢ C 6 = J 2⊗ [ 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 ] C 7 = J 2 ⊗ [ 0 0 1 0 0 0 0 - 1 10 0 0 0 - 1 0 0 ] C 8 = J 2 ⊗ [ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ] ⁢ ⁢ J 2= [ 0 - 1 1 0 ] , ( 8 ) with h ~ 8 ⁢ k - 7 = { vec ( [ ( H k R ) T , ( Hk I ) T ] T ) , for ⁢ ⁢ quasi ⁢ - ⁢ orthogonal , vec ( [ ( H k R ) T , 0 N ×2 , ( H k I ) T , 0 N × 2 ] T ) , for ⁢ ⁢ Alamouti . ⁢ ( 9 )Finally, for a single transmit antenna BS, defining {tilde over(C)}_(i)=(I_(N){circle around (×)}C_(i)), we have that{tilde over (H)} _(k) =[{tilde over (h)} _(8k−7) , . . . , {tilde over(h)} _(8k) ]={tilde over (C)} ₁ {tilde over (h)} _(8k−7) ,−{tilde over(C)} ₂ {tilde over (h)} _(8k−7) , {tilde over (C)} ₃ {tilde over (h)}_(8k−7) ,−{tilde over (C)} ₄ {tilde over (h)} _(8k−7) , {tilde over (C)}₅ {tilde over (h)} _(8k−7) , {tilde over (C)} ₆ {tilde over (h)} _(8k−7), {tilde over (C)} ₇ {tilde over (h)} _(8k−7) , {tilde over (C)} ₈{tilde over (h)} _(8k−7)]  (10)and

${\overset{\sim}{h}}_{{8k} - 7} = {{{vec}\left( \left\lbrack {\left( H_{k}^{R} \right)^{T},0_{N \times 3},\left( H_{k}^{I} \right)^{T},0_{N \times 3}} \right\rbrack^{T} \right)}.}$Further, we let {tilde over (W)}

diag{w₁, . . . , w_(K)}{circle around (×)}I₈ and define{tilde over (H)} _(k)

[{tilde over (H)}_(k+1), . . . , {tilde over (H)}_(K)],  (11){tilde over (W)} _(k)

diag{w_(k+1), . . . , w_(K)}{circle around (×)}I₈.  (12)

2.2. Group Decoders

We consider the decoding of a frame received over T=4J, J≧1 consecutivesymbol intervals, where over a block of 4 consecutive symbol intervals(or four consecutive tones in an OFDMA system) we obtain a model of theform in (6). We first consider the group MMSE decision-feedback decoder(GM-DFD), where the user decodes and cancels the signals of as manyinterfering BSs as necessary before decoding the desired signal. We thenconsider the group MMSE decoder (GMD) where the user only decodes thedesired BS after suppressing the signals of all the interfering BSs.

2.2.1. Group MMSE Decision-Feedback Decoder (GM-DFD)

For ease of exposition, we assume that BS k is the desired one and thatthe BSs are decoded in the increasing order of their indices, i.e., BS 1is decoded first, BS 2 is decoded second and so on. Note that no attemptis made to decode the signals of BSs k+1 to K.

The soft statistics for the first BS over 4 consecutive symbolintervals, denoted by {tilde over (r)}₁, are obtained as,{tilde over (r)} ₁ ={tilde over (F)} _(1{tilde over (y)}) ={tilde over(F)} ₁ {tilde over (H)} ₁ {tilde over (x)} ₁ +ũ ₁,  (13)where {tilde over (F)}₁ denotes the MMSE filter for BS 1 and is givenby, {tilde over (F)}₁={tilde over (H)}₁ ^(T)(σ²I+{tilde over (H)} ₁{tilde over (W)} ₁ {tilde over (H)} ₁ ^(T))⁻¹ and ũ₁={tilde over (F)}₁{tilde over (H)} ₁ {tilde over (x)} ₁ +{tilde over (F)}₁{tilde over(v)}₁ and note that{tilde over (Σ)}₁

E[ũ ₁ ũ ₁ ^(T) ]={tilde over (F)} ₁ {tilde over (H)} ₁ ={tilde over (H)}₁ ^(T)(σ² I+{tilde over (H)} ₁ {tilde over (W)} ₁ {tilde over (H)} ₁^(T))⁻¹ {tilde over (H)} ₁.  (14)

To decode BS 1, ũ₁ is assumed to be a colored Gaussian noise vector withthe covariance in (14). Under this assumption, in the case when no outercode is employed by BS 1, the decoder obtains a hard decision {tildeover (x)}₁, using the maximum-likelihood (ML) rule over the model in(13). On the other hand, if an outer code is employed by BS 1soft-outputs for each coded bit in {tilde over (x)}₁ are obtained usingthe soft-output MIMO demodulator over the model in (13), which are thenfed to a decoder. The decoded codeword is re-encoded and modulated toobtain the decision vectors {{tilde over (x)}₁} over the frame ofduration 4J symbol intervals. In either case, the decision vectors{{tilde over (x)}₁} are fed back before decoding the subsequent BSs. Inparticular, the soft statistics for the desired k^(th) BS, are obtainedas,

$\begin{matrix}{{{\overset{\sim}{r}}_{k} = {{\overset{\sim}{F}}_{k}\left( {\overset{\sim}{y} - {\sum\limits_{j = 1}^{k - 1}\;{{\overset{\sim}{H}}_{j}{\hat{x}}_{j}}}} \right)}},} & (15)\end{matrix}$where {tilde over (F)}_(k) denotes the MMSE filter for BS k and is givenby, {tilde over (F)}_(k)={tilde over (H)}_(k) ^(T)(σ²I+{tilde over (H)}_(k) {tilde over (W)} _(k) {tilde over (H)} _(k) ^(T))⁻¹. The decoderfor the BS k is restricted to be a function of {{tilde over (r)}_(k)}and obtains the decisions {{circumflex over (x)}_(k)} in a similarmanner after assuming perfect feedback and assuming the additive noiseplus interference to be Gaussian. Note that the choice of decoding BSs 1to k−1 prior to BS k was arbitrary. In the sequel we will address theissue of choosing an appropriate ordered subset of interferers to decodeprior to the desired signal.

2.2.2. Group MMSE Decoder (GMD)

We assume that BS 1 is the desired one so that only BS 1 is decodedafter suppressing the interference from BSs 2 to K. The soft statisticsfor the desired BS are exactly {tilde over (r)}₁ given in (13). Notethat the MMSE filter for BS 1 can be written as {tilde over (F)}₁={tildeover (H)}₁ ^(T)({tilde over (R)} ₁ )⁻¹ where {tilde over (R)} ₁=σ²I+{tilde over (H)} ₁ {tilde over (W)} ₁ {tilde over (H)} ₁ ^(T),denotes the covariance matrix of the noise plus interference. Thus toimplement this decoder we only need estimates of the channel matrixcorresponding to the desired signal and the covariance matrix. Also, theuser need not be aware of the inner code employed by any of theinterfering BSs. In this work we assume perfect estimation of thechannel as well as the covariance matrices.

Inspecting the models in (13) and (15), we see that the complexity ofimplementing the ML detection (demodulation) for the k^(th) BS (underthe assumption of perfect feedback in case of GM-DFD) directly dependson the structure of the matrix {tilde over (F)}_(k) {tilde over(H)}_(k). Ideally, the matrix {tilde over (F)}_(k) {tilde over (H)}_(k)should be diagonal which results in a linear complexity and if most ofthe off-diagonal elements of {tilde over (F)}_(k) {tilde over (H)}_(k)are zero, then the cost of implementing the detector (demodulator) issignificantly reduced. Henceforth, for notational convenience we willabsorb the matrix {tilde over (W)} in the matrix {tilde over (H)}, i.e.,we will denote the matrix {tilde over (H)}{tilde over (W)} by {tildeover (H)}.

3. Decoupling Property

In this section we prove a property which results in significantly lowerdemodulation complexity. Note that the matrices defined in (8) have thefollowing properties:C _(l) ^(T) =C _(l),

ε {1,3}, C _(l) ^(T) =−C _(l),

ε {1, . . . , 8}\{1,3}, C _(l) ^(T) C _(l) =I, ∀

  (16)In addition they also satisfy the ones given in Table 1, shown below,

TABLE I PROPERTIES OF {C_(i)} C₁ C₂ C₃ C₄ C₅ C₆ C₇ C₈ C₁ ^(T) C₁ C₂ C₃C₄ C₅ C₆ C₇ C₈ C₂ ^(T) −C₂ C₁ −C₄ C₃ C₆ −C₅ C₈ −C₇ C₃ ^(T) C₃ C₄ C₁ C₂C₇ C₈ C₅ C₆ C₄ ^(T) −C₄ C₃ −C₂ C₁ C₈ −C₇ C₆ −C₅ C₅ ^(T) −C₅ −C₆ −C₇ −C₈C₁ C₂ C₃ C₄ C₆ ^(T) −C₆ C₅ −C₈ C₇ −C₂ C₁ −C₄ C₃ C₇ ^(T) −C₇ −C₈ −C₅ −C₆C₃ C₄ C₁ C₂ C₈ ^(T) −C₈ C₇ −C₆ C₅ −C₄ C₃ −C₂ C₁where the matrix in the (i, j)^(th) position is obtained as the resultof C_(i) ^(T)C_(j). Thus, the set of matrices ∪_(i=1) ⁸{±C_(i)} isclosed under matrix multiplication and the transpose operation. We offerthe following theorem.

Theorem 1. Consider the decoding of the k^(th) BS. We have that{tilde over (H)} _(k) ^(T)(σ² I+{tilde over (H)} _(k) {tilde over (H)}_(k) ^(T))⁻¹ {tilde over (H)} _(k)=α_(k) C ₁+β_(k) C ₃,  (17)for some real-valued scalars α_(k), β_(k). Note that α_(k), β_(k) dependon {tilde over (H)}_(k) and {tilde over (H)} _(k) but for notationalconvenience we do not explicitly indicate the dependence.Proof. To prove the theorem, without loss of generality we will onlyconsider decoding of the first BS. We first note that

$\begin{matrix}{{{{\sigma^{2}I} + {{\overset{\sim}{H}}_{\overset{\_}{1}}{\overset{\sim}{H}}_{\overset{\_}{1}}^{T}}} = {\sum\limits_{i = 1}^{8}\;{\left( {I_{N} \otimes C_{i}} \right){\overset{\sim}{A}\left( {I_{N} \otimes C_{i}^{T}} \right)}}}},} & (18)\end{matrix}$where

${\overset{\sim}{A}{\sigma^{2}/8}I} + {\sum\limits_{k = 1}^{K}\;{{\overset{\sim}{h}}_{{8k} - 7}{{\overset{\sim}{h}}_{{8k} - 7}^{T}.}}}$Let {tilde over (B)}

(σ²I+{tilde over (H)} ₁ {tilde over (H)} ₁ ^(T))⁻¹ and note that {tildeover (B)}>0. Using the properties of the matrices {C_(i)} in (16) andTable 1, it is readily verified that

${\left( {I_{N} \otimes C_{i}} \right){\overset{\sim}{B}\left( {I_{N} \otimes C_{i}^{T}} \right)}} = {\left( {\left( {I_{N} \otimes C_{i}} \right)\left( {\sum\limits_{i = 1}^{8}\;{\left( {I_{N} \otimes C_{i}} \right){\overset{\sim}{A}\left( {I_{N} \otimes C_{i}^{T}} \right)}}} \right)\left( {I_{N} \otimes C_{i}^{T}} \right)} \right)^{- 1} = {\overset{\sim}{B}.}}$As a consequence we can expand {tilde over (B)} as

$\begin{matrix}{\overset{\sim}{B} = {\sum\limits_{i = 1}^{8}\;{\left( {I_{N} \otimes C_{i}} \right)\left( {\overset{\sim}{B}/8} \right){\left( {I_{N} \otimes C_{i}^{T}} \right).}}}} & \text{(19)}\end{matrix}$Next, invoking the properties of the matrices {C_(i)} and using the factthat {tilde over (B)}={tilde over (B)}^(T), it can be seen that thematrix

${\left( {I_{N} \otimes C_{k}^{T}} \right)\left( {\sum\limits_{i = 1}^{8}\;{\left( {I_{N} \otimes C_{i}} \right)\left( {\hat{B}/8} \right)\left( {I_{N} \otimes C_{i}^{T}} \right)}} \right)\left( {I_{N} \otimes C_{j}} \right)},$where 1≦k, j≦8, is identical to {tilde over (B)} when k=j, is identicalwhen (k, j) or (j, k)ε {(1,3), (2,4), (5,7), (6,8)} and is skewsymmetric otherwise. The desired property in (17) directly follows fromthese facts.

Note that Theorem 1 guarantees the quasi-orthogonality property evenafter interference suppression. In particular, the important point whichcan be inferred from Theorem 1 is that the joint detection(demodulation) of four complex QAM symbols (or eight PAM symbols) issplit into four smaller joint detection (demodulation) problemsinvolving a pair of PAM symbols each. Thus with four M-QAM complexsymbols the complexity is reduced from

(M⁴) to

(M). Furthermore, specializing Theorem 1 to the case when the desired BS(say BS k) employs the quasi-orthogonal design and there are nointerferers, we see that{tilde over (H)} _(k) ^(T) {tilde over (H)} _(k)=α_(k) C ₁+β_(k) C₃.  (20)(20) implies that maximum likelihood decoding complexity of thequasi-orthogonal design is

(M) instead of the more pessimistic

(M²) claimed by the original contribution. We note that a differentquasi-orthogonal design referred to as the minimum decoding complexityquasi-orthogonal design, was proposed for a point-to-point MIMO systemin the prior art, which was shown to have an ML decoding complexity of

(M).Finally, it can be inferred from the sequel that β_(k)=0 in (17), whenno BS in {k, k+1, . . . , K} employs the quasi orthogonal design.4. Efficient Inverse Computation

In this section we utilize the structure of the covariance matrix {tildeover (R)}

σ²I+{tilde over (H)}{tilde over (H)}^(T) to efficiently compute itsinverse. Consequently, the complexity involved in computing the MMSEfilters is significantly reduced. Let {tilde over (S)}={tilde over(R)}⁻¹ From (18) and (19), it follows that we can expand both {tildeover (R)}, {tilde over (S)} as

$\begin{matrix}{{\overset{\sim}{R} = \begin{bmatrix}{\sum\limits_{i = 1}^{8}\;{C_{i}P_{11}C_{i}^{T}}} & \cdots & {\sum\limits_{i = 1}^{8}\;{C_{i}P_{1N}C_{i}^{T}}} \\\vdots & \cdots & \vdots \\{\sum\limits_{i = 1}^{8}\;{C_{i}P_{N\; 1}C_{i}^{T}}} & \cdots & {\sum\limits_{i = 1}^{8}\;{C_{i}P_{NN}C_{i}^{T}}}\end{bmatrix}}{{\overset{\sim}{S} = \begin{bmatrix}{\sum\limits_{i = 1}^{8}\;{C_{i}Q_{11}C_{i}^{T}}} & \cdots & {\sum\limits_{i = 1}^{8}\;{C_{i}Q_{1N}C_{i}^{T}}} \\\vdots & \cdots & \vdots \\{\sum\limits_{i = 1}^{8}\;{C_{i}Q_{N\; 1}C_{i}^{T}}} & \cdots & {\sum\limits_{i = 1}^{8}\;{C_{i}Q_{NN}C_{i}^{T}}}\end{bmatrix}},}} & (21)\end{matrix}$where {P_(ij), Q_(ij)}_(i,j=1) ^(N) are 8×8 matrices such thatP_(ji)=P_(ij) ^(T), Q_(ji)=Q_(ij) ^(T), 1≦i, j≦N.  (22)

The inverse {tilde over (S)} can be computed recursively starting fromthe bottom-right sub-matrix of {tilde over (R)} using the followinginverse formula for block partitioned matrices

$\begin{matrix}{\begin{bmatrix}E & F \\G & H\end{bmatrix}^{- 1} = \begin{bmatrix}\left( {E - {{FH}^{- 1}G}} \right)^{- 1} & {{- \left( {E - {{FH}^{- 1}G}} \right)^{- 1}}{FH}^{- 1}} \\{{- H^{- 1}}{G\left( {E - {{FH}^{- 1}G}} \right)}^{- 1}} & {H^{- 1} + {H^{- 1}{G\left( {E - {{FH}^{- 1}G}} \right)}^{- 1}{FH}^{- 1}}}\end{bmatrix}} & (23)\end{matrix}$The following properties ensure that the computations involved indetermining {tilde over (S)} are dramatically reduced.First, note that the 8×8 sub-matrices in (21) belong to the set ofmatrices

$\begin{matrix}{\left\{ {{??}\overset{\Delta}{=}{\sum\limits_{i = 1}^{8}\;{C_{i}A\;{C_{i}^{T}:{A \in {IR}^{8 \times 8}}}}}} \right\}.} & (24)\end{matrix}$

It is evident that

is closed under the transpose operation. Utilizing the structure of thematrices {C_(i)} in (8), after some algebra it can be shown that the set

can also be written as

$\begin{matrix}{{\underset{\_}{??}\overset{\Delta}{=}\left\{ {\sum\limits_{i = 1}^{8}\;{b_{i}{S_{i}:{\left\lbrack {b_{1},\ldots\mspace{11mu},b_{8}} \right\rbrack^{T} \in {IR}^{8}}}}} \right\}},} & (25)\end{matrix}$where S₁=I₈, S₅=J₂{circle around (×)}I₄, S₃=C₃ and

$\begin{matrix}{{{S_{2} = {\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix} \otimes \begin{bmatrix}0 & 1 & 0 & 0 \\{- 1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & {- 1} & 0\end{bmatrix}}}S_{4} = {\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix} \otimes \begin{bmatrix}0 & 0 & 0 & 1 \\0 & 0 & {- 1} & 0 \\0 & 1 & 0 & 0 \\{- 1} & 0 & 0 & 0\end{bmatrix}}}\;{S_{6} = {\begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix} \otimes \begin{bmatrix}0 & {- 1} & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & {- 1} \\0 & 0 & 1 & 0\end{bmatrix}}}\mspace{20mu}{S_{7} = {{{J_{2} \otimes \begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{bmatrix}}\mspace{25mu} S_{8}} = {\begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix} \otimes {\begin{bmatrix}0 & 0 & 0 & 1 \\0 & 0 & {- 1} & 0 \\0 & 1 & 0 & 0 \\{- 1} & 0 & 0 & 0\end{bmatrix}.}}}}} & (26)\end{matrix}$

It is readily seen that the set

in (25) is a matrix group under matrix addition and note that any matrixB ε

is parametrized by eight scalars. The matrices {S_(i)} have thefollowing properties.S _(l) ^(T) =S _(l),

ε {1,3}, S _(l) ^(T) =−S _(l),

ε {1, . . . , 8}\{1,3}, S _(l) ^(T) S _(l)=, ∀

  (27)in addition to the ones given in Table II, shown below.

TABLE II PROPERTIES OF {S_(i)} S₁ S₂ S₃ S₄ S₅ S₆ S₇ S₈ S₁ ^(T) S₁ S₂ S₃S₄ S₅ S₆ S₇ S₈ S₂ ^(T) −S₂ S₁ −S₄ S₃ −S₆ S₅ S₈ −S₇ S₃ ^(T) S₃ S₄ S₁ S₂S₇ −S₈ S₅ −S₆ S₄ ^(T) −S₄ S₃ −S₂ S₁ S₈ S₇ −S₆ −S₅ S₅ ^(T) −S₅ S₆ −S₇ −S₈S₁ −S₂ S₃ S₄ S₆ ^(T) −S₆ −S₅ S₈ −S₇ S₂ S₁ S₄ −S₃ S₇ ^(T) −S₇ −S₈ −S₅ S₆S₃ −S₄ S₁ S₂ S₈ ^(T) −S₈ S₇ S₆ S₅ −S₄ −S₃ −S₂ S₁Using these properties it can be verified that the set {±S_(i)}_(i=1) ⁸is closed under matrix multiplication and the transpose operation. Thefollowing lemma provides useful properties of the set

.

$\begin{matrix}{{Lemma}\mspace{14mu} 1.} & \; \\{A,{B \in {\mspace{14mu} A\; B} \in}} & (28) \\{A = {{A^{T} \in {\mspace{11mu} A}} = {{{a_{1}I_{8}} + {a_{2}S_{3}}} = {{a_{1}I_{8}} + {a_{2}C_{3}}}}}} & (29) \\{A = {\left. {{{{{a_{1}I_{8}} + {a_{2}S_{3}}}\&}\;{A}} \neq 0}\Rightarrow A^{- 1} \right. = {{\frac{a_{1}}{a_{1}^{2} - a_{2}^{2}}I_{8}} - {\frac{a_{2}}{a_{1}^{2} - a_{2}^{2}}S_{3}}}}} & (30)\end{matrix}$for some scalars a₁, a₂ and

$\begin{matrix}{{{\sum\limits_{i = 1}^{8}{C_{i}{BC}_{i}^{T}}} = {{{b_{1}I_{8}} + {b_{2}S_{3}}} = {{b_{1}I_{8}} + {b_{2}C_{3}}}}},{{\forall B} = {B^{T} \in {IR}^{8 \times 8}}}} & (31) \\{{Q \in {Q\; Q^{T}}} = {{q_{1}I_{8}} + {q_{2}C_{3}}}} & (32)\end{matrix}$for some scalars b₁, b₂, q₁, q₂.Proof The facts in (28) and (29) follow directly by using the alternateform of

in (25) along with the properties of {S_(i)}. (30) follows after somesimple algebra whereas (31) follows from (29) upon using the definitionof

in (24). Finally (32) follows from (28) and (29) after recalling thatthe set

is closed under the transpose operation.Thus for any A, B ε

the entire 8×8 matrix AB can be determined by only computing any one ofits rows (or columns). The set

is not a matrix group since it contains singular matrices. However theset of all nonsingular matrices in

forms a matrix group as shown by the following lemma.

Lemma 2. If A ε

such that |A|≠0 then A⁻¹ε

The set of all non-singular matrices in

denoted by

forms a matrix group under matrix multiplication and is given by

$\begin{matrix}{= \left\{ {{{{\sum\limits_{i = 1}^{8}{b_{i}{S_{i}:{\left\lbrack {b_{1},\ldots\;,b_{8}} \right\rbrack^{T} \in {IR}^{8}}}}}\&}{\sum\limits_{i = 1}^{8}b_{i}^{2}}} \neq {{\pm 2}\left( {{b_{1}b_{3}} + {b_{2}b_{4}} + {b_{5}b_{7}} - {b_{6}b_{8}}} \right)}} \right\}} & (33)\end{matrix}$Proof. Consider any non-singular A ε

so that A⁻¹ exists. We can use the definition of

in (24) to expand A as

$\sum\limits_{j = 1}^{8}\;{C_{j}{QC}_{j}^{T}}$for some Q ε IR^(8×8). Consequently

$A^{- 1} = {\left( {\sum\limits_{j = 1}^{8}\;{C_{j}{QC}_{j}^{T}}} \right)^{- 1}.}$Next, as done in the proof of Theorem 1, using the properties of {C_(i)}we can show that

${C_{i}A^{- 1}C_{i}^{T}} = {\left( {{C_{i}\left( {\sum\limits_{j = 1}^{8}\;{C_{j}{QC}_{j}^{T}}} \right)}C_{i}^{T}} \right)^{- 1} = {A^{- 1}.}}$Thus, we have that

$\begin{matrix}{{A^{- 1} = {\sum\limits_{j = 1}^{8}{{C_{J}\left( {A^{- 1}/8} \right)}C_{J}^{T}}}},} & (34)\end{matrix}$so that A⁻¹ε

Next, using the alternate form of

in (25) we must have that A=Σ_(i=1) ⁸a_(i)S_(i), for some {a_(i)}. Sincethe non-singular A ε

we must have that AA^(T) ε

and note that|A|≠0

|AA^(T)|>0.  (35)Invoking the property in (32), after some algebra we see that

$\begin{matrix}{{A\; A^{T}} = {{\sum\limits_{i = 1}^{8}{a_{i}^{2}I_{8}}} + {2\left( {{a_{1}a_{3}} + {a_{2}a_{4}} + {a_{5}a_{7}} - {a_{6}a_{8}}} \right){C_{3}.}}}} & (36)\end{matrix}$

Then it can be verified that

$\begin{matrix}{{{A\; A^{T}}} = {\left( {\left( {\sum\limits_{i = 1}^{8}a_{i}^{2}} \right)^{2} - {4\left( {{a_{1}a_{3}} + {a_{2}a_{4}} + {a_{5}a_{7}} - {a_{6}a_{8}}} \right)^{2}}} \right)^{4}.}} & (37)\end{matrix}$From (35) and (37), we see that the set

is precisely the set of all non-singular matrices in

Since this set includes the identity matrix, is closed under matrixmultiplication and inversion, it is a matrix group under matrixmultiplication.

Lemma 2 is helpful in computing the inverses of the principalsub-matrices of {tilde over (R)}. Note that since {tilde over (R)}>0,all its principal sub-matrices are also positive-definite and hencenon-singular. Then, to compute the inverse of any A ε

we can use Lemma 2 to conclude that A⁻¹ε

so that we need to determine only the eight scalars which parametrizeA⁻¹. As mentioned before, in this work we assume that a perfect estimateof the covariance matrix {tilde over (R)} is available. In practice thecovariance matrix {tilde over (R)} must be estimated from the receivedsamples. We have observed that the Ledoit and Wolf's (LW) estimator [10]works well in practice. For completeness we provide the LW estimator.Let {{tilde over (y)}_(n)}_(n=1) ^(S) be the S vectors which areobtained from samples received over 4S consecutive symbol intervals overwhich the effective channel matrix {tilde over (H)} in (6) is constant.These samples could also be received over consecutive tones and symbolsin an OFDMA system. Then the LW estimate {tilde over ({circumflex over(R)} is given by

$\begin{matrix}{{\hat{\overset{\sim}{R}} = {{\left( {1 - \rho} \right)\hat{Q}} + {{\mu\rho}\; I}}},{{{where}\mspace{14mu}\hat{Q}} = {\frac{1}{S}{\sum\limits_{n = 1}^{S}{{\overset{\sim}{y}}_{n}{\overset{\sim}{y}}_{n}^{T}\mspace{14mu}{and}}}}}} & (38) \\{{\rho = {\min\left\{ {\frac{\sum\limits_{n = 1}^{S}{{{{\overset{\sim}{y}}_{n}{\overset{\sim}{y}}_{n}^{T}} - \hat{Q}}}_{F}^{2}}{S^{2}{{\hat{Q} - {\mu\; I}}}_{F}^{2}},1} \right\}}}{{{and}\mspace{14mu}\mu} = {\frac{t\;{r\left( \hat{Q} \right)}}{8N}.}}} & (39)\end{matrix}$5. GM-DFD: Decoding Order

It is well known that the performance of decision feedback decoders isstrongly dependent on the order of decoding. Here however, we are onlyconcerned with the error probability obtained for the signal of thedesired (serving) BS. Note that the GM-DFD results in identicalperformance for the desired BS for any two decoding orders where theordered sets of BSs decoded prior to the desired one, respectively, areidentical. Using this observation, we see that the optimal albeitbrute-force method to decode the signal of the desired BS using theGM-DFD would be to sequentially examine

$\sum\limits_{i = 0}^{K - 1}{{i!}\begin{pmatrix}{K - 1} \\i\end{pmatrix}}$possible decoding orders, where the ordered sets of BSs decoded prior tothe desired one are distinct for any two decoding orders, and pick thefirst one where the signal of desired BS is correctly decoded, which inpractice can be determined via a cyclic redundancy check (CRC). Althoughthe optimal method does not examine all K! possible decoding orders, itcan be prohibitively complex. We propose an process which determines theBSs (along with the corresponding decoding order) that must be decodedbefore the desired one. The remaining BSs are not decoded.

The challenge in designing such a process is that while canceling acorrectly decoded interferer clearly aids the decoding of the desiredsignal, the subtraction of even one erroneously decoded signal canresult in a decoding error for the desired signal. Before providing theprocess we need to establish some notation. We let

={1, . . . , K} denote the set of BSs and let k denote the index of thedesired BS. Let R_(j), 1≦j≦K denote the rate (in bits per channel use)at which the BS j transmits. Also, we let π denote any ordered subset ofK having k as its last element. For a given π, we let π(1) denote itsfirst element, which is also the index of the BS decoded first by theGM-DFD, π(2) denote its second element, which is also the index of theBS decoded second by the GM-DFD and so on. Finally let |π| denote thecardinality of π and let Q denote the set of all possible such π.

Let us define m({tilde over (H)}, j, S) to be a metric whose value isproportional to the chance of successful decoding of BS j in thepresence of interference from BSs in the set S. A large value of themetric implies a high chance of successfully decoding BS j. Further, weadopt the convention that m({tilde over (H)}, φ, S)=∞∀S, since no erroris possible in decoding the empty set. Define {tilde over(H)}_(S)=[{tilde over (H)}_(j)]_(jεS). Let I({tilde over (H)}, j, S)denote an achievable rate (in bits per channel use) obtained post MMSEfiltering for BS j in the presence of interference from BSs in the set Sand note that

$\begin{matrix}\begin{matrix}{{I\left( {\overset{\sim}{H},j,S} \right)} = {\frac{1}{2}\log{{I_{8} + {{{\overset{\sim}{H}}_{j}^{T}\left( {{\sigma^{2}I} + {\overset{\sim}{H_{S}}{\overset{\sim}{H}}_{S}^{T}}} \right)}^{- 1}\overset{\sim}{H_{J}}}}}}} \\{{= {2\;{\log\left( {\left( {1 + \alpha_{j,S}} \right)^{2} - \beta_{j,S}^{2}} \right)}}},}\end{matrix} & (40)\end{matrix}$where the second equality follows upon using (17). In this work wesuggest the following three examples for m({tilde over (H)}, j, S)

$\begin{matrix}{{{m\left( {\overset{\sim}{H},j,S} \right)} = {{I\left( {\overset{\sim}{H},j,S} \right)} - R_{j}}},} & (41) \\{{{m\left( {\overset{\sim}{H},j,S} \right)} = {{I\left( {\overset{\sim}{H},j,S} \right)}/R_{j}}},{and}} & (42) \\\begin{matrix}{{m\left( {\overset{\sim}{H},j,S} \right)} = {\max\limits_{\rho \in {\lbrack{0,1}\rbrack}}{\rho\left( {{\frac{1}{2}\log{{I_{8} + {\frac{1}{1 + \rho}{{\overset{\sim}{H}}_{j}^{T}\left( {\sigma^{2} + {\overset{\sim}{H_{S}}{\overset{\sim}{H}}_{S}^{T}}} \right)}^{- 1}\overset{\sim}{H_{J}}}}}} - R_{j}} \right)}}} \\{= {\max\limits_{\rho \in {\lbrack{0,1}\rbrack}}{{\rho\left( {{2{\log\left( {\left( {1 + \frac{\alpha_{j,S}}{1 + \rho}} \right)^{2} - \frac{\beta_{j,S}^{2}}{\left( {1 + \rho} \right)^{2}}} \right)}} - R_{j}} \right)}.}}}\end{matrix} & (43)\end{matrix}$

Note that the metric in (43) is the Gaussian random coding errorexponent obtained after assuming BSs in the set S to be Gaussianinterferers. All three metrics are applicable to general non-symmetricsystems where the BSs may transmit at different rates. It can be readilyverified that all the three metrics given above also satisfy thefollowing simple factm({tilde over (H)}, j, S)≧m({tilde over (H)}, j,

), ∀S ⊂

⊂ K.  (44)Now, for a given π ε Q, the metric m(H,k,

\∪_(j=1) ^(|π|)π(j)) indicates the decoding reliability of the desiredsignal assuming perfect feedback from previously decoded signals,whereas min_(1≦j≦|π|−1) m({tilde over (H)}, π(j),

\∪_(i=1) ^(j)π(i)) can be used to measure the quality of the fed-backdecisions. Thus a sensible metric to select π is

$\begin{matrix}{{f\left( {H,\pi} \right)}\overset{\Delta}{=}{\min\limits_{1 \leq j \leq {\pi }}{{m\left( {\overset{\sim}{H},{\pi(j)},{\backslash{\bigcup_{i = 1}^{j}{\pi(i)}}}} \right)}.}}} & (45)\end{matrix}$

We are now ready to present our process.

-   -   1. Initialize: S={1, . . . , K} and {circumflex over (π)}=φ.    -   2. Among all BS indices j ε S, select the one having the highest        value of the metric m({tilde over (H)}, j, S\j) and denote it by        ĵ.    -   3. Update S=S\ĵ and {circumflex over (π)}={{circumflex over        (π)}, ĵ}.    -   4. If ĵ=k then stop else go to Step 2.        The proposed greedy process is optimal in the following sense.

Theorem 2. The process has the following optimality.

$\begin{matrix}{\hat{\pi} = {\arg\;{\max\limits_{\pi \in \underset{\_}{Q}}{{f\left( {\overset{\sim}{H},\pi} \right)}.}}}} & (46)\end{matrix}$Proof. Let π^((i)) be any other valid ordered partition in Q such thatits first i elements are identical to those of {circumflex over (π)}.Construct another ordered partition π^((i+1)) as follows:π^((i+1))(j)=π^((i))(j)={circumflex over (π)}(j), 1≦j≦i,π^((i+1))(i+1)={circumflex over (π)}(i+1),π^((i+1))(j+1)=π^((i))(j)\{circumflex over (π)}(i+1), i+1≦j≦|π ^((i))|&{circumflex over (π)}(i+1)≠k  (47)Note that π^((i+1)) ε Q. Now, to prove optimality it is enough to showthatf({tilde over (H)}, π^((i+1)))≧f({tilde over (H)}, π^((i))).  (48)To show (48) we first note thatm({tilde over (H)}, π^((i+1))(j), K\∪ _(q=1) ^(j)π^((i+1))(q))=m({tildeover (H)}, π ^((i))(j), K\∪ _(q=1) ^(j)π^((i))(q)), 1≦j≦i.  (49)Since the greedy process selects the element (BS) with the highestmetric at any stage, we have thatm({tilde over (H)}, π^((i+1))(i+1),\∪_(q=1) ^(i+1)π^((i+1))(q))≧m({tildeover (H)}, π ^((i))(i+1),\∪_(q=1) ^(i+1)π^((i))(q)).  (50)If {circumflex over (π)}(i+1) equals k then (49) and (50) prove thetheorem, else using (85) we see thatm({tilde over (H)}, π ^((i+1))(j+1),\∪_(q=1)^(j+1)π^((i+1))(q))≧m({tilde over (H)}, π ^((i))(j),\∪_(q=1)^(j)π^((i))(q)), i+1≦j≦|π ^((i))|.  (51)From (51), (50) and (49) we have the desired result.

The following remarks are now in order.

-   -   The metrics in (41)-to-(43) are computed assuming Gaussian input        alphabet and Gaussian interference. We can exploit the available        modulation information by computing these metrics for the exact        alphabets (constellations) used by all BSs but this makes the        metric computation quite involved. We can also compute the        metric m({tilde over (H)}, j,) by assuming the BSs in the set of        interferers S to be Gaussian interferers but using the actual        alphabet for the BS j, which results in a simpler metric        computation. In this work, we use the first (and simplest)        option by computing the metrics as in (82)-to-(84). Moreover,        the resulting decoding orders are shown in the sequel to perform        quite well with finite alphabets and practical outer codes.    -   A simple way to achieve the performance of the optimal GM-DFD        with a lower average complexity, is to first examine the        decoding order suggested by the greedy process and only in the        case the desired BS is decoded erroneously, to sequentially        examine the remaining

${\sum\limits_{i = 0}^{K - 1}\;{{i!}\begin{pmatrix}{K - 1} \\i\end{pmatrix}}} - 1$decoding orders.

-   -   Note that when f({tilde over (H)}, {circumflex over (π)})—where        {circumflex over (π)} is the order determined by the greedy        rule—is negative, less than 1 and equal to 0 when m({tilde over        (H)}, j, S) is computed according to (41), (42) and (43),        respectively, we can infer that with high probability at least        one BS will be decoded in error. In particular, suppose we use        the metric in (41). Then an error will occur (with high        probability) for the desired BS k even after perfect        cancellation of the previous BSs if m({tilde over (H)}, k,        \∪_(j=1) ^(|{circumflex over (π)}|){circumflex over (π)}(j))<0.        On the other hand, when m({tilde over (H)}, k,        \∪_(j=1) ^(|{circumflex over (π)}|){circumflex over (π)}(j))>0        but min_(1≦j≦|{circumflex over (π)}|−1) m({tilde over (H)},        {circumflex over (π)}(j),        \∪_(i=1) ^(j){circumflex over (π)}(i))<0, we can infer that the        decoding of the desired BS will be affected (with high        probability) by error propagation from BSs decoded previously.        Unfortunately, it is hard to capture the effect of error        propagation precisely and we have observed that the assumption        that error propagation always leads to a decoding error for the        desired BS is quite pessimistic.        6. Special Cases

In this section a lower complexity GMD is obtained at the cost ofpotential performance degradation by considering only two consecutivesymbol intervals when designing the group MMSE filter. Further, when nointerfering BS employs the quasi-orthogonal design no loss of optimalityis incurred. Similarly, when none of the BSs employ the quasi-orthogonaldesign, without loss of optimality we can design the GM-DFD byconsidering only two consecutive symbol intervals.

In this case, the 2×N channel output received over two consecutivesymbol intervals can be written as (1). As before, the transmittedmatrix X can be partitioned as X=[X₁, . . . , X_(K)] but where

$\begin{matrix}{{X_{k} = \begin{bmatrix}x_{k,1} & x_{k,2} \\{- x_{k,2}^{\dagger}} & x_{k,1}^{\dagger}\end{bmatrix}},} & (52)\end{matrix}$when the k^(th) BS employs the Alamouti design andX_(k)=[x_(k,1) x_(k,2)]^(T),  (53)when the k^(th) BS has only one transmit antenna. Note that over twoconsecutive symbol intervals, an interfering BS employing thequasi-orthogonal design is equivalent to two dual transmit antenna BSs,each employing the Alamouti design. Then we can obtain a linear model ofthe form in (6), where {tilde over (x)}=[{tilde over (x)}₁ ^(T), . . . ,{tilde over (x)}_(K) ^(T)]^(T) and {tilde over (x)}_(k)=[x_(k,1) ^(R),x_(k,2) ^(R), x_(k,3) ^(I), x_(k,2) ^(I)]^(T) with {tilde over(H)}=[{tilde over (H)}₁, . . . , {tilde over (H)}_(K)]=[{tilde over(h)}₁, . . . , {tilde over (h)}_(4K)]. The matrix {tilde over (H)}_(k)corresponding to a BS employing the Alamouti design can be expanded as{tilde over (H)} _(k) [{tilde over (h)} _(4k−3) , . . . , {tilde over(h)} _(4k) ]=[{tilde over (h)} _(4k−3), (I _(N) {circle around (×)}D₁){tilde over (h)} _(4k−3), (I _(N) {circle around (×)}D ₂){tilde over(h)} _(4k−3), (I _(N) {circle around (×)}D ₃){tilde over (h)}_(4k−3)],  (54)with {tilde over (h)}_(4k−3)=vec([(H_(k) ^(R))^(T), (H_(k)^(I))^(T)]^(T)), whereas that corresponding to a single transmit antennaBS can be expanded as{tilde over (H)} _(k) [{tilde over (h)} _(4k−3) , . . . , {tilde over(h)} _(4k) ]=[{tilde over (h)} _(4k−3),−(I _(N) {circle around (×)}D₁){tilde over (h)} _(4k−3), (I _(N) {circle around (×)}D ₂){tilde over(h)} _(4k−3), (i _(N) {circle around (×)}d ₃){tilde over (h)}_(4k−3)],  (55)with {tilde over (h)}_(4k−3)=vec([(H_(k) ^(R))^(T),0_(N×1), (H_(k)^(I))^(T),0_(N×1)]^(T)). The matrices D₁, D₂, D₃ are given by

$\begin{matrix}{{D_{1}\overset{\Delta}{=}\begin{bmatrix}0 & 1 & 0 & 0 \\{- 1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & {- 1} & 0\end{bmatrix}}\mspace{14mu}{D_{2}\overset{\Delta}{=}\begin{bmatrix}0 & 0 & {- 1} & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & {- 1} & 0 & 0\end{bmatrix}}{D_{3}\overset{\Delta}{=}{\begin{bmatrix}0 & 0 & 0 & {- 1} \\0 & 0 & {- 1} & 0 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0\end{bmatrix}.}}} & (56)\end{matrix}$Note that the matrices defined in (56) have the following properties:D_(l) ^(T)=−D_(l), D_(l) ^(T)D_(l)=I, 1≦l≦3  (57)D₂ ^(T)D₁=−D₃, D₂ ^(T)D₃=D₁, D₁ ^(T)D₃=−D₂.Using the properties given in (57), we can prove the following theoremin a manner similar to that of Theorem 1. The proof is skipped forbrevity.

Theorem 3. Consider the decoding of the k^(th) BS. We have that{tilde over (H)} _(k) ^(T)(σ² I+{tilde over (H)} _(k) {tilde over (H)}_(k) ^(T))⁻¹ {tilde over (H)} _(k)=α_(k) I ₄.  (58)Let

σ²I+{tilde over (H)}{tilde over (H)}^(T) denote a sample covariancematrix obtained by considering two consecutive symbol intervals. Define_(k=[)2k−1,2k,4N+2k−1,4N+2k], 1≦k≦2N and e=[e₁, . . . , e_(2N)] and letM denote the permutation matrix obtained by permuting the rows of I_(8N)according to e. Then, it can be verified that the matrices in (7) and(10), corresponding to Alamouti and single antenna BSs (over four symbolintervals), are equal (up to a column permutation) to M(I₂{circle around(×)}{tilde over (H)}_(k)), where {tilde over (H)}_(k) is given by (54)and (55), respectively. Consequently, the covariance matrix {tilde over(R)} in (21) is equal to M(I₂{circle around (×)}Ũ)M^(T), when noquasi-orthogonal BSs are present, so that {tilde over (R)}⁻¹=M(I₂{circlearound (×)}Ũ⁻¹)M^(T). Moreover, it can be shown that the decouplingproperty also holds when the desired BS employs the quasi-orthogonaldesign and the filters are designed by considering two consecutivesymbol intervals. Note that designing the MMSE filter by considering twoconsecutive symbol intervals implicitly assumes that no quasi-orthogonalinterferers are present, so the demodulation is done accordingly.

Next, we consider the efficient computation of the inverse {tilde over(V)}=Ũ⁻¹. Letting D₀=I₄, analogous to (18) and (19), it can be shownthat we can expand both Ũ, {tilde over (V)} as

$\overset{\sim}{U} = \begin{bmatrix}{\sum\limits_{i = 0}^{3}{D_{i}P_{11}D_{i}^{T}}} & \ldots & {\sum\limits_{i = 0}^{3}{D_{i}P_{1N}D_{i}^{T}}} \\\vdots & \ldots & \vdots \\{\sum\limits_{i = 0}^{3}{D_{i}P_{N\; 1}D_{i}^{T}}} & \ldots & {\sum\limits_{i = 0}^{3}{D_{i}P_{N\; N}D_{i}^{T}}}\end{bmatrix}$ ${\overset{\sim}{V} = \begin{bmatrix}{\sum\limits_{i = 0}^{3}{D_{i}Q_{11}D_{i}^{T}}} & \ldots & {\sum\limits_{i = 0}^{3}{D_{i}Q_{1N}D_{i}^{T}}} \\\vdots & \ldots & \vdots \\{\sum\limits_{i = 0}^{3}{D_{i}Q_{N\; 1}D_{i}^{T}}} & \ldots & {\sum\limits_{i = 0}^{3}{D_{i}Q_{N\; N}D_{i}^{T}}}\end{bmatrix}},$where {P_(ij), Q_(ij)})_(i,j=1) ^(N) are now 4×4 matrices satisfying(22).The inverse computation can be done recursively using the formula in(23). The following observations greatly reduce the number ofcomputation involved.

First, utilizing the properties of the matrices {D_(i)} in (57), we canshow that the set

$\begin{matrix}\begin{matrix}{\underset{\_}{Q\overset{\Delta}{=}}\left\{ {\sum\limits_{i = 0}^{3}{D_{i}{{AD}_{i}^{T}:{A \in {IR}^{4 \times 4}}}}} \right\}} \\{{= \left\{ {\sum\limits_{i = 0}^{3}{b_{i}{T_{i}:{\left\lbrack {b_{0},\ldots\mspace{11mu},b_{3}} \right\rbrack \in {I\; R^{4}}}}}} \right\}},}\end{matrix} & (59)\end{matrix}$where T₀=I₄, and

$T_{1} = \begin{bmatrix}0 & 1 & 0 & 0 \\{- 1} & 0 & 0 & 0 \\0 & 0 & 0 & {- 1} \\0 & 0 & 1 & 0\end{bmatrix}$ ${T_{2} = \begin{bmatrix}0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1} \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{bmatrix}}\;$ $T_{3} = {\begin{bmatrix}0 & 0 & {0 - 1} & 0 \\0 & 0 & 1 & 0 \\0 & {- 1} & 0 & 0 \\1 & 0 & 0 & 0\end{bmatrix}.}$Thus Q is closed under the transpose operation and any matrix B ε Q isparametrized by four scalars. The matrices {T_(i)} have the followingproperties:T_(l) ^(T)=−T_(l), T_(l) ^(T)T_(l)=I, 1≦l≦3  (60)T₂ ^(T)T₁=T₃, T₂ ^(T)T₃=−T₁, T₁ ^(T)T₃=T₂.Using these properties it can be verified that the set {±T_(i)}_(i=1) ⁸is closed under matrix multiplication and the transpose operation. Thefollowing two lemmas provide useful properties of the set Q. The proofsare similar to those of the previous two lemmas and hence are skippedfor brevity.

$\begin{matrix}{{Lemma}\mspace{14mu} 3.} & \; \\{{A,\left. {B \in \underset{\_}{Q}}\Rightarrow{{A\; B} \in \underset{\_}{Q}} \right.}{{A = {\left. {A^{T} \in \underset{\_}{Q}}\Rightarrow A \right. = {a_{1}I_{4}}}},}} & (61)\end{matrix}$for some scalar a₁ and

$\begin{matrix}{{{\sum\limits_{i = 0}^{3}{D_{i}{BD}_{i}^{T}}} = {{b_{1}I_{4}\mspace{14mu}{\forall B}} = {B^{T} \in {IR}^{4 \times 4}}}}{{\left. {Q \in \underset{\_}{Q}}\Rightarrow{Q\; Q^{T}} \right. = {q_{1}I_{4}}},}} & (62)\end{matrix}$for some scalars b₁, q₁.Thus for any A, B εQ, the entire 4×4 matrix AB can be determined by onlycomputing any one of its rows (or columns). Further, the set of allnonsingular matrices in Q forms a matrix group under matrixmultiplication and is given by,

$\underset{\_}{\overset{\sim}{Q}} = {\left\{ {\sum\limits_{i = 0}^{3}{b_{i}{T_{i}:{\left\lbrack {b_{0},\ldots\mspace{14mu},b_{3}} \right\rbrack^{T} \in {{IR}^{4}\backslash O}}}}} \right\}.}$

The present invention has been shown and described in what areconsidered to be the most practical and preferred embodiments. It isanticipated, however, that departures may be made therefrom and thatobvious modifications will be implemented by those skilled in the art.It will be appreciated that those skilled in the art will be able todevise numerous arrangements and variations which, not explicitly shownor described herein, embody the principles of the invention and arewithin their spirit and scope.

1. A method for decoding in a wireless downlink channel, where alldominant transmitting sources use inner codes from a particular set,comprising the steps of: estimating a channel matrix seen from eachdominant transmitter source in response to a pilot or preamble signaltransmitted by each such source; converting each estimated channelmatrix into an effective channel matrix responsive to the inner code ofthe corresponding transmitting source; obtaining received signalobservations in a linear equivalent form whose output is an equivalentof the received observations and in which the effective channel matrixcorresponding to each dominant transmitting source inherits thestructure of its inner code; i) determining an order for processing eachof the transmitting sources comprising a decodability metric responsiveto a decoupling property so that the number of operations involved in acomputing are reduced; ii) computing a filter for each transmittingsource that will be decoded; iii) demodulating and decoding eachtransmitting source responsive to the determined order from step i)assuming perfect cancellation of signals of preceding or previouslydecoded transmitting sources; and iv) re-encoding a decoded message ofeach transmitting source, except a transmitting source decoded last,responsive to a modulation and coding scheme employed by thetransmitting source and the corresponding effective channel matrix andsubtracting it from the received observations in the equivalent linearform.
 2. The method of claim 1, wherein the transmitter source can havedistributed or non co-located physical transmit antennas.
 3. The methodof claim 2, wherein the transmitter source comprises being formed by twoor more transmitter sources which pool their transmit antennas andcooperatively transmit a signal to a destination receiver.
 4. The methodof claim 1, wherein the step of determining the order for processingeach of the transmitting sources is done in a greedy manner, where ateach stage the transmitter source maximizing a decodability metric isselected, until all sources of interest have been selected, at eachstage the value of the decodability metric computed for a transmittersource is proportional to the chance of successful decoding of thattransmitter source in the presence of interference from the remainingun-decoded sources.
 5. The method of claim 1, wherein the step ofdemodulating the transmitter source is also responsive to a decouplingproperty according to the following relationship H_(k) ^(T)(σ²I+H _(k) H_(k) ^(T))⁻¹H_(k)=α_(k)C₁+β_(k)C₃, where H_(k) is the effective channelmatrix corresponding to a desired transmitted signal source (with indexk), H _(k) is the effective channel matrix corresponding to un-decodeddominant interfering sources and α_(k), β_(k) are scalars which dependon H_(k), H _(k) and σ², where σ² is a noise variance which can includean average received power from other interfering sources in addition toa thermal noise variance C₁ that is an 8 times 8 identity matrix and C₃is a particular matrix.
 6. A method for decoding in a wireless downlinkchannel, where all dominant transmitting sources use inner codes from aparticular set, comprising the steps of: estimating a channel matrixseen from each dominant transmitter source in response to a pilot orpreamble signal transmitted by each such source; converting eachestimated channel matrix into an effective channel matrix responsive tothe inner code of the corresponding transmitting source; obtainingreceived signal observations in a linear equivalent form whose output isan equivalent of the received observations and in which the effectivechannel matrix corresponding to each dominant transmitting sourceinherits the structure of its inner code; i) determining an order forprocessing each of the transmitting sources; ii) computing a filter foreach transmitting source that will be decoded; iii) demodulating anddecoding each transmitting source responsive to the determined orderfrom step i) assuming perfect cancellation of signals of preceding orpreviously decoded transmitting sources, said demodulating beingresponsive to a decoupling property in which a gain matrix obtained postinterference suppression in a linear model after assuming perfectcancellation of previously decoded sources, guarantees aquasi-orthogonality property; and iv) re-encoding a decoded message ofeach transmitting source, except a transmitting source decoded last,responsive to a modulation and coding scheme employed by thetransmitting source and the corresponding effective channel matrix andsubtracting it from the received observations in the equivalent linearform.
 7. The method of claim 1, wherein the step of determining an orderfor processing each of the transmitting sources comprises computing adecodability metric utilizing a structure of a covariance matrix andproperties of its 8×8 sub-matrices.
 8. The method of claim 1, whereinthe covariance matrix of the noise plus signals transmitted bytransmitter sources for which either the channel estimates or the innercodes or the modulation and coding schemes are unknown, is estimatedusing output vectors in a linear model as sample input vectors for anestimator.
 9. The method of claim 8, wherein the output of said linearmodel is an equivalent of the received observations in which theeffective channel matrix corresponding to each transmitter sourceinherits the structure of its inner code.
 10. The method of claim 8,wherein step of estimating the covariance matrix is followed byprocessing the covariance matrix estimate obtained from the estimator toensure that the processed matrix has a structure for an efficientinverse computation.
 11. The method of claim 6, wherein thequasi-orthogonality property comprises that the joint detection ordemodulation of four complex QAM symbols (or eight PAM symbols) is splitinto four smaller joint detection (demodulation) problems involving apair of PAM symbols each, thereby with four M-QAM complex symbols ademodulation complexity being reduced from

(M⁴) to

(M).
 12. A method for decoding in a wireless downlink channel, where alldominant transmitting sources use inner codes from a particular set,comprising the steps of: estimating a channel matrix seen from eachdominant transmitter source in response to a pilot or preamble signaltransmitted by each such source; converting each estimated channelmatrix into an effective channel matrix responsive to the inner code ofthe corresponding transmitting source; obtaining received signalobservations in a linear equivalent form whose output is an equivalentof the received observations and in which the effective channel matrixcorresponding to each dominant transmitting source inherits thestructure of its inner code; i) determining an order for processing eachof the transmitting sources; ii) computing a filter for eachtransmitting source that will be decoded; iii) demodulating and decodingeach transmitting source responsive to the determined order from step i)assuming perfect cancellation of signals of preceding or previouslydecoded transmitting sources; and iv) re-encoding a decoded message ofeach transmitting source, except a transmitting source decoded last,responsive to a modulation and coding scheme employed by thetransmitting source and the corresponding effective channel matrix andsubtracting it from the received observations in the equivalent linearform; wherein the channel estimates or the inner codes or the modulationand coding schemes are not known for some of the transmitter sources,the signals transmitted by which are consequently only treated asinterference and deemed un-decodable.